Problem 26 - College Algebra with Modeling and Visualization

Question

Find the circumference and area of the circle. Approximate each value to the nearest tenth when appropriate. \(r=5 z\)

Step-by-Step Solution

Answer

Circumference is approximately \(31.4z\) and area is approximately \(78.5z^2\).

Step 1: Understand the Problem

The problem gives us a circle with a radius of \( r = 5z \). We are asked to find two things: the circumference and the area of the circle. We will use the formula for circumference \( C = 2\pi r \) and for area \( A = \pi r^2 \).

Step 2: Calculate the Circumference

To find the circumference, use the formula \( C = 2\pi r \). Substitute \( r = 5z \) into the formula, so we have:\[C = 2\pi (5z) = 10\pi z\]This is the exact expression for the circumference of the circle.

Step 3: Calculate the Area

To find the area, use the formula \( A = \pi r^2 \). Substitute \( r = 5z \) into the formula, so we have:\[A = \pi (5z)^2 = \pi (25z^2) = 25\pi z^2\]This is the exact expression for the area of the circle.

Step 4: Approximate the Results

To approximate the circumference and area to the nearest tenth, use \(\pi \approx 3.14\).For the circumference:\[10\pi z \approx 10 \times 3.14 \times z = 31.4z\]For the area:\[25\pi z^2 \approx 25 \times 3.14 \times z^2 = 78.5z^2\]

Key Concepts

Circumference

The circumference of a circle is the distance around it. It is one of the fundamental properties of a circle and tells us how far one would travel if they went around the circle once. To find the circumference, we use the formula: \[ C = 2\pi r \] This formula shows that the circumference depends on the radius of the circle and the constant \( \pi \), which is approximately \( 3.14159 \). In the given problem, the radius is provided as \( 5z \). By substituting this value into the formula, we get: \[ C = 2\pi (5z) = 10\pi z \] This tells us the exact expression for the circumference based on the variable \( z \). To approximate it, substitute \( \pi \) with \( 3.14 \): \[ C \approx 31.4z \] So, if \( z \) had a specific value, you could easily calculate the circumference by multiplying \( 31.4 \) by that value of \( z \). This insight helps us understand how the circumference scales with different values of the radius.

Area of a Circle

The area of a circle is the measure of the space inside the circle. It’s essential for understanding how much surface the circle covers. The area can be calculated using the formula: \[ A = \pi r^2 \] For our exercise, the circle's radius is \( r = 5z \), so when we substitute \( r \) with \( 5z \), the formula becomes: \[ A = \pi (5z)^2 = 25\pi z^2 \] This simplified form gives us a formula that relates the area directly with the square of \( z \). To approximate this value, we use \( \pi \approx 3.14 \), leading to: \[ A \approx 78.5z^2 \] This approximation helps visualize how the area changes with variations in the variable \( z \). The squared relationship means that as the radius grows, the area increases quite rapidly, showcasing the power of the square in areas of circles.

Radius

The radius of a circle is a line segment from the center of the circle to any point on its boundary. It's a critical component in both geometric and real-world applications because it directly impacts both the circumference and area of the circle. In mathematical terms, the radius is often used as the basis for calculations: - The circumference can be found using the radius in the formula \( C = 2\pi r \). - The area is calculated using the radius in \( A = \pi r^2 \). In our example, the radius is given as \( 5z \). The use of a variable like \( z \) implies that the radius can change, which affects all other calculations involving the circle. By understanding the role of the radius, you gain insight into how changes in size directly translate to changes in crust coverage (area) and border length (circumference). Knowing the radius also helps in identifying circle properties quickly and efficiently. Even a small increase in radius can lead to significant increases in both circumference and area.